The quadratic growth conjecture for colored radial orderings

Let $g_\mathrm{col}(n)$ denote the relevant extremal quantity for colored radial orderings of bichromatic sets of $2n$ points. Quadratic colored-ordering conjecture.

$$g_\mathrm{col}(n)=\Theta(n^2).$$

The paper notes that bichromatic sets can have $\Theta(n^4)$ colored radial orderings, while other sets have only $\Theta(n^2)$, and reports only an $\Omega(n)$ lower bound before making this conjecture.